Question

The amount of pollution content added in air in a city to $x$ diesel vehicles is given by $P\left(x\right)=0.005x^3+0.02x^2+30x$. Find the marginal increase in pollution content when $3$ diesel vehicles are added.

Answer 1

$P\left(x\right)=0.005x^3+0.02x^2+30x$
Differentiating w.r.t to x we get,
$P^{\prime }\left(x\right)=3\left(0.005\right)x^2+2\left(0.02\right)x+30$
$P^{\prime }\left(3\right)=3\times 0.005(3{)}^2+2(0.02\left)\right(3)+30$
$P^{\prime }\left(3\right)=30.255$